3. Reprimitivization, Physical Sets, Partonomies, Levels, and Sem-linking

Since the three bonding forces vary so dramatically in strength, timing, and scale, there are domains in which each can be analyzed almost entirely in isolation. The domain of the strong force is nuclear physics, the domain of electromagnetism is chemistry, and the domain of gravity is cosmology. The bonds within each of these depend entirely on the force of the domain, and it’s useful to imagine the entities being bonded as primitives in their own right within that domain, in a process I call reprimitivization, giving a new pile of Dust. In nuclear physics, no reprimitivization is necessary because quarks are in fact primitives. The electromagnetic Dust for chemistry is electrons and nuclei, and only the latter need to be reprimitivized. In cosmology, the new primitives are stars, planets, black holes, moons, and other celestial bodies. Each of these domains has frayed edges—the frays between nuclear physics and chemistry being in contexts like nuclear fission and neutron star collapse, the frays between chemistry and cosmology in contexts like planetary formation and comet tails.

Within each domain, pairs or groups of primitives that are bonded can be bonded further, usually with bonds that operate at a different scale of strength, distance, or timing. In the nucleus, triplets of quarks bond through the strong force into nucleons, i.e. protons and neutrons, and anywhere from 2 to well over 100 nucleons bond together through the weaker residual strong force into entire nuclei. In chemistry, electrons bond to one nucleus to form neutral atoms and charged ions, to two or a few1 nuclei to form covalent molecules or polyatomic ions, or to indefinitely many to form metals. Atoms bonded to other atoms through covalent bonds can form molecules in a range from minimal clumps like carbon monoxide or water, to indefinitely long chains like polymers and plastics, to indefinitely flat sheets like those in talc and graphite, to indefinitely bulky 3D matrices like quartz and diamond. Ions bond to form salts. Molecules bond through Van der Waals forces to form bulk materials. In cosmology, primitives are bonded either in bundle of orbits like in a solar system, or in a mostly unstructured cloud of gravitational interaction like in a star cluster or a galaxy.

The act of bonding creates a relationship between two or more primitives. To characterize such relationships, let’s take a short detour through mathematics to acquire some relevant vocabulary. The “set,” a foundational notion in math, is defined as a collection of distinct objects.2 Examples of sets include the even numbers, the odd numbers, the multiples of 13, negative numbers. Of course, sets aren’t restricted to collections of numbers, so there are more examples like collections of linear functions, finite groups, differentiable manifolds, and the rest of the moonshine that mathematicians study. The notation for sets consists of two curly braces as bookends, with some stuff in between, separated by commas. Let’s say we want a set that contains the integers between 4 and 10, excluding 4 and 10. Then the braces with the “stuff in between” is: {5, 6, 7, 8, 9}. Five elements—pretty easy. There’s also some special notation for the set containing no elements, the empty set: Ø, which could otherwise by written as {}.

Now things get interesting. We can use sets to make sets! Let’s make the set {5, 6, {7, 8}, 9}. Here we have a set containing 5, 6, a set containing 7 and 8, and 9. The set has 4 elements, one of which is a set containing 2 elements. It may be confusing why it has 4 and not 5. In addition to the conventional notion of  “element,” let’s entertain the notion of a “mathematical primitive” and think of it as a mathematical object that isn’t a set.3 Then we can assert that though the set contains 4 elements, it contains 5 mathematical primitives. How about the set {5, 6, Ø, 9}? This set also contains 4 elements, but curiously contains only 3 mathematical primitives! There’s a sense in which the empty set creates something from nothing.4 More vividly, we can ask if {Ø} = Ø? Certainly not! {Ø}, on the left, isn’t empty; it in fact contains Ø. Ø, on the right, is empty; it contains nothing. So we then observe that while Ø contains nothing, it itself is not nothing. Finally, we can observe that {5, 6, 7, 8, 9} does not equal {5, 6, {7, 8}, 9} because the elements don’t match.

An important notion in applied math is the “tree,” which is a specific type of graph. A graph is a mathematical object constructed from points or “vertices” together with the lines or “edges” that connect them.5 A tree is a graph that is connected and has no loops in it; that is, there is always exactly one path along edges and vertices to get from any one vertex to any other without backtracking. The file structure of all computers as far as I know is a tree, which you can see in most file viewers by expanding all folders and subfolders if you’ve ever had enough time to waste. The leaves of a tree are those vertices that are connected to only one edge and are in a sense on the “tips” of the tree. In a computer file system, that would be all of the individual files in whichever folders. If you consider mathematical primitives as leaves, then every set has a corresponding tree, where the primitives and sets are vertices and the edges connect sets to their contents, either other sets or primitives. Such a tree can also be called a nested hierarchy, or just a hierarchy.


Figure 1. A few sets and their trees.

But what does this have to do with physical bonding? The bonding patterns of each force within its domain can be regarded as sets. A nucleus, for instance, is a set of 1-100+ sets of 3 quarks. The nucleus of helium-4 as an example would be {{qu1, qu2, qd1}, {qu1, qu2, qd1}, {qu1, qd1, qd2}, {qu1, qd1, qd2}}. A molecule, say water, would be {nucH1, e1, e2, e3, e4, {e5, nucO1, e6} e7, e8, e9, e10, nucH2}, although a conventional Lewis diagram would be clearer.6 The solar System would be {Sun, {Mercury}, {Venus}, {Earth, {Moon}}, {Mars, {Phobos, Deimos}}, {asteroids}, {Jupiter, {Metis, Adrastea, Amalthea, Thebe, Io, Europa, etc.}}, etc.}. Each of these would be a physical set, and more specifically a strong nuclear set, an electromagnetic set, and a gravitational set, respectively. The tree or nested hierarchy of each physical set is called its “partonomy,” since it breaks down the parthood relationships within the set,7 and each nesting creates a level, with the Dust at the lowest level and each reprimitivization on its own level.

Reprimitivization is quite interesting because it’s fundamentally fraudulent—there is only one real Dust heap—while still being very useful. Reprimitivization, however, is merely a special case of the more general fraudulent process of treating a collection of primitives or a collection of collections just like an individual primitive. Notice the ease with which you can talk about protons and electrons in the same sentence. Further, reprimitivization operates at levels where the responsibility of a particular bonding force for the collecting is always very clear, but often we will have collections where there is no such clarity. As in the first chapter, a power strip is a collection of sockets and wires and other parts, but which of the bonding forces is responsible for keeping them together? Which of the bonding forces is responsible for keeping a tree together? An answer isn’t terribly difficult to deduce (hint: electromagnetism), but it would be difficult to talk about the attraction and repulsion of Dust in the elevator-pitch version of the answer.

I call that process sem-linking, for reasons that will be clear later. Reprimitivization sem-links nucleons into nuclei as chemical primitives and enormous quantities of chemicals into stars and planets as cosmological primitives, but sem-linking itself covers even more ground, taking us from three quarks to a proton and from half a dozen sockets to a power strip and from branches and leaves to a tree. Basically, every node in a partonomy that is not a leaf is the sem-linking of some of the nodes in the level below it. Further, the physical sets constructed by sem-linking will have interactions of their own, which of course are predicated on the interactions of the Dust, the four fundamental forces of the Standard Model.

Footnotes

1. As in benzene or phenyl groups, for instance

2. In a set, the objects must be distinct. If they’re not, you have a multiset. If a set or multiset is ordered, then you have a tuple. If a tuple only contains numbers, then you have a vector. If you order numbers along two or more dimensions, then you have a matrix or a tensor.

3. Also known as an “ur-element.”

4. This is less fanciful than it sounds. Mathematical systems can construct the numbers and beyond from just the empty set, like the standard Zermelo-Fraenkel axiomatization of set theory. Integers would not be “mathematical primitives” in this case, but rather carefully constructed sets in their own right. The mathematical primitive there is just the empty set and the nothing it contains. See G. Spencer-Brown’s Laws of Form.

5. A graph also means a picture used to plot information, often in the Cartesian plane, but that is separate meaning of the word.

6. But it would also hide the two non-valence electrons in the first shell around the oxygen nucleus.

7. Also called a “meronomy,” but the Special Composition Question can wait.

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